Why does mesh not follow bone perfectly?
It looks like it's a bit messy in your vertex groups and bone weights, for example if I select a group like Chest (which is duplicated by the way), but also some other bones, then press the Select button, I can see that the arms are part of these groups, maybe not enough so I can see it in Weight Paint mode, but enough to have influence on it. So maybe check all your groups, remove these vertices from these groups, or delete some bones and/or weight paint again...